Geometry, Topology, and Dynamics in Negative Curvature
Geometry, Topology, and Dynamics in Negative Curvature

Author: C. S. Aravinda

Publisher: Cambridge University Press

Published: 2016-01-21

Total Pages: 378

ISBN-13: 1316539180

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The ICM 2010 satellite conference 'Geometry, Topology and Dynamics in Negative Curvature' afforded an excellent opportunity to discuss various aspects of this fascinating interdisciplinary subject in which methods and techniques from geometry, topology, and dynamics often interact in novel and interesting ways. Containing ten survey articles written by some of the leading experts in the field, this proceedings volume provides an overview of important recent developments relating to negative curvature. Topics covered include homogeneous dynamics, harmonic manifolds, the Atiyah Conjecture, counting circles and arcs, and hyperbolic buildings. Each author pays particular attention to the expository aspects, making the book particularly useful for graduate students and mathematicians interested in transitioning from other areas via the common theme of negative curvature.

Geometry, Topology, and Dynamics in Negative Curvature
Geometry, Topology, and Dynamics in Negative Curvature

Author: C. S. Aravinda

Publisher: Cambridge University Press

Published: 2016-01-27

Total Pages: 0

ISBN-13: 9781316540909

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The ICM 2010 satellite conference 'Geometry, Topology and Dynamics in Negative Curvature' afforded an excellent opportunity to discuss various aspects of this fascinating interdisciplinary subject in which methods and techniques from geometry, topology, and dynamics often interact in novel and interesting ways. Containing ten survey articles written by some of the leading experts in the field, this proceedings volume provides an overview of important recent developments relating to negative curvature. Topics covered include homogeneous dynamics, harmonic manifolds, the Atiyah Conjecture, counting circles and arcs, and hyperbolic buildings. Each author pays particular attention to the expository aspects, making the book particularly useful for graduate students and mathematicians interested in transitioning from other areas via the common theme of negative curvature.

Analytic and Probabilistic Approaches to Dynamics in Negative Curvature
Analytic and Probabilistic Approaches to Dynamics in Negative Curvature

Author: Françoise Dal'Bo

Publisher: Springer

Published: 2014-07-17

Total Pages: 148

ISBN-13: 3319048074

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The work consists of two introductory courses, developing different points of view on the study of the asymptotic behaviour of the geodesic flow, namely: the probabilistic approach via martingales and mixing (by Stéphane Le Borgne); the semi-classical approach, by operator theory and resonances (by Frédéric Faure and Masato Tsujii). The contributions aim to give a self-contained introduction to the ideas behind the three different approaches to the investigation of hyperbolic dynamics. The first contribution focus on the convergence towards a Gaussian law of suitably normalized ergodic sums (Central Limit Theorem). The second one deals with Transfer Operators and the structure of their spectrum (Ruelle-Pollicott resonances), explaining the relation with the asymptotics of time correlation function and the periodic orbits of the dynamics.

Geometry, Topology, and Dynamics
Geometry, Topology, and Dynamics

Author: François Lalonde

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 158

ISBN-13: 082180877X

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This is a collection of papers written by leading experts. They are all clear, comprehensive, and origianl. The volume covers a complete range of exciting and new developments in symplectic and contact geometries.

Ergodic Theory and Negative Curvature
Ergodic Theory and Negative Curvature

Author: Boris Hasselblatt

Publisher: Springer

Published: 2017-12-15

Total Pages: 334

ISBN-13: 3319430599

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Focussing on the mathematics related to the recent proof of ergodicity of the (Weil–Petersson) geodesic flow on a nonpositively curved space whose points are negatively curved metrics on surfaces, this book provides a broad introduction to an important current area of research. It offers original textbook-level material suitable for introductory or advanced courses as well as deep insights into the state of the art of the field, making it useful as a reference and for self-study. The first chapters introduce hyperbolic dynamics, ergodic theory and geodesic and horocycle flows, and include an English translation of Hadamard's original proof of the Stable-Manifold Theorem. An outline of the strategy, motivation and context behind the ergodicity proof is followed by a careful exposition of it (using the Hopf argument) and of the pertinent context of Teichmüller theory. Finally, some complementary lectures describe the deep connections between geodesic flows in negative curvature and Diophantine approximation.

Wigner-Type Theorems for Hilbert Grassmannians
Wigner-Type Theorems for Hilbert Grassmannians

Author: Mark Pankov

Publisher: Cambridge University Press

Published: 2020-01-16

Total Pages: 155

ISBN-13: 1108848397

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Wigner's theorem is a fundamental part of the mathematical formulation of quantum mechanics. The theorem characterizes unitary and anti-unitary operators as symmetries of quantum mechanical systems, and is a key result when relating preserver problems to quantum mechanics. At the heart of this book is a geometric approach to Wigner-type theorems, unifying both classical and more recent results. Readers are initiated in a wide range of topics from geometric transformations of Grassmannians to lattices of closed subspaces, before moving on to a discussion of applications. An introduction to all the key aspects of the basic theory is included as are plenty of examples, making this book a useful resource for beginning graduate students and non-experts, as well as a helpful reference for specialist researchers.

(Co)end Calculus
(Co)end Calculus

Author: Fosco Loregian

Publisher: Cambridge University Press

Published: 2021-07-22

Total Pages: 331

ISBN-13: 1108746128

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This easy-to-cite handbook gives the first systematic treatment of the (co)end calculus in category theory and its applications.

Discrete Quantum Walks on Graphs and Digraphs
Discrete Quantum Walks on Graphs and Digraphs

Author: Chris Godsil

Publisher: Cambridge University Press

Published: 2023-01-12

Total Pages: 152

ISBN-13: 1009261703

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Discrete quantum walks are quantum analogues of classical random walks. They are an important tool in quantum computing and a number of algorithms can be viewed as discrete quantum walks, in particular Grover's search algorithm. These walks are constructed on an underlying graph, and so there is a relation between properties of walks and properties of the graph. This book studies the mathematical problems that arise from this connection, and the different classes of walks that arise. Written at a level suitable for graduate students in mathematics, the only prerequisites are linear algebra and basic graph theory; no prior knowledge of physics is required. The text serves as an introduction to this important and rapidly developing area for mathematicians and as a detailed reference for computer scientists and physicists working on quantum information theory.

New Directions in Locally Compact Groups
New Directions in Locally Compact Groups

Author: Pierre-Emmanuel Caprace

Publisher: Cambridge University Press

Published: 2018-02-08

Total Pages: 367

ISBN-13: 1108349544

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This collection of expository articles by a range of established experts and newer researchers provides an overview of the recent developments in the theory of locally compact groups. It includes introductory articles on totally disconnected locally compact groups, profinite groups, p-adic Lie groups and the metric geometry of locally compact groups. Concrete examples, including groups acting on trees and Neretin groups, are discussed in detail. An outline of the emerging structure theory of locally compact groups beyond the connected case is presented through three complementary approaches: Willis' theory of the scale function, global decompositions by means of subnormal series, and the local approach relying on the structure lattice. An introduction to lattices, invariant random subgroups and L2-invariants, and a brief account of the Burger–Mozes construction of simple lattices are also included. A final chapter collects various problems suggesting future research directions.

Partial Differential Equations in Fluid Mechanics
Partial Differential Equations in Fluid Mechanics

Author: Charles L. Fefferman

Publisher: Cambridge University Press

Published: 2018-09-27

Total Pages: 339

ISBN-13: 1108573592

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The Euler and Navier–Stokes equations are the fundamental mathematical models of fluid mechanics, and their study remains central in the modern theory of partial differential equations. This volume of articles, derived from the workshop 'PDEs in Fluid Mechanics' held at the University of Warwick in 2016, serves to consolidate, survey and further advance research in this area. It contains reviews of recent progress and classical results, as well as cutting-edge research articles. Topics include Onsager's conjecture for energy conservation in the Euler equations, weak-strong uniqueness in fluid models and several chapters address the Navier–Stokes equations directly; in particular, a retelling of Leray's formative 1934 paper in modern mathematical language. The book also covers more general PDE methods with applications in fluid mechanics and beyond. This collection will serve as a helpful overview of current research for graduate students new to the area and for more established researchers.